Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How I Learned to Love Algebraic Geometry

johncarlosbaez.wordpress.com

Post reply on HN
Print view View original

31–40 of 56 posts

Re: How I Learned to Love Algebraic Geometry

#31

Post by obastani »

NiftyWalrus wrote: post #30
obastani wrote: post #27

Earlier quoted context omitted.

The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures

What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.

I don't know myself, but here are some possible answers:

https://mathoverflow.net/questions/114034/learning-path-for-...

Edit: Here is another overview that seems good:

https://www.math.ucdavis.edu/~osserman/math/pcm.pdf

Re: How I Learned to Love Algebraic Geometry

#32

Post by heinrichf »

kevinventullo wrote: post #5

Algebraic Geometry is a powerful tool of number theory because much of it works over any field. It allows one to translate geometric intuition (algebraic geometry over the complex numbers) into a more algebraic environment (finite, p-adic, or number fields). Going back further, algebraic geometry over the complex numbers was shown in the early 20th century to be in many ways equivalent to more classical analytic geom…

More than over any field, over any ring. For example, that allows you to look at families of curves or talk about reduction modulo p (if you do number theory) very nicely/naturally.

A mathoverflow thread with cool uses of schemes: https://mathoverflow.net/questions/59071/what-elementary-pro...

Re: How I Learned to Love Algebraic Geometry

#33

Post by MaysonL »

NiftyWalrus wrote: post #30
obastani wrote: post #27

Earlier quoted context omitted.

The "Background and history" section of the following article gives a very high-level idea of Grothendieck's contribution: https://en.wikipedia.org/wiki/Weil_conjectures

What is the modern path to study this area now? I’m sure it’s better understood now and one wouldn’t have to follow the historical approach to study the same concepts.

Take a look at the stacks project: https://stacks.math.columbia.edu

A cross between a wiki and a collaborative textbook.

Re: How I Learned to Love Algebraic Geometry

#34

Post by danharaj »

eggy wrote: post #23

Can geometric algebra be used in algebraic geometry? I understand geometric algebra is useful in physics involving differential geometry and calculus, and that algebraic geometry is heavy on the algebra of polynomials and a complete field of study in mathematics, whereas geometric algebra is more like an object (Clifford algebra, etc.).

Geometric algebra is about the algebra of a widget called a grassmanian, which figures very prominently in algebraic geometry.

Re: How I Learned to Love Algebraic Geometry

#35

Post by MaysonL »

ginnungagap wrote: post #13
maaaats wrote: post #11

What are some good resources for grasping or learning more about algebraic geometry?

You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the firs…

Vakil's notes have an interesting epigram from Grothendieck:

I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!…

A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet finally it surrounds the resistant substance.

— A. Grothendieck

Re: How I Learned to Love Algebraic Geometry

#36

Post by eggy »

danharaj wrote: post #34
eggy wrote: post #23

Can geometric algebra be used in algebraic geometry? I understand geometric algebra is useful in physics involving differential geometry and calculus, and that algebraic geometry is heavy on the algebra of polynomials and a complete field of study in mathematics, whereas geometric algebra is more like an object (Clifford algebra, etc.).

Geometric algebra is about the algebra of a widget called a grassmanian, which figures very prominently in algebraic geometry.

I have been playing with John Browne's Grassmann Algebra in Mathetmatica for my geometric algebra studies, but this is the first I've encountered the Grassmannian. Thanks, I'll look into it.

Re: How I Learned to Love Algebraic Geometry

#37

Post by notduncansmith »

umutisik wrote: post #4

“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?

Complex numbers is one area where sine is very relevant.

Fascinating article about the geometry of complex numbers: https://acko.net/blog/how-to-fold-a-julia-fractal/

Re: How I Learned to Love Algebraic Geometry

#38

Post by ginnungagap »

MaysonL wrote: post #35
ginnungagap wrote: post #13

Earlier quoted context omitted.

You'll need a good background in commutative algebra to learn algebraic geometry, if you have that and want to see the modern approach, schemes and everything else that Grothendieck did, I suggest Vakil's notes, they're freely available from his homepage. (Disclaimer: I only studied the first 15 chapters and the one on Kähler differentials which should be the 21st, but I suppose the second half is as good as the firs…

Vakil's notes have an interesting epigram from Grothendieck: I can illustrate the … approach with the … image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe…

Vakil seems very fond of that quote, the full title of his notes is "The Rising Sea - Foundations of Algebraic Geometry"!

Re: How I Learned to Love Algebraic Geometry

#40

Post by wolfgke »

kevinventullo wrote: post #6
umutisik wrote: post #4

“Mathematicians study curves described by all sorts of equations – but sines, cosines and other fancy functions are only a distraction from the fundamental mysteries of the relation between geometry and algebra.” Is this statement backed up by theorems or is it being made because there are just more proven theorems in algebraic geometry and there is not much work on solution sets to more general equations?

I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring. Going back to the 1800's, in the theory of Riemann surfaces (one-dimensional complex manifolds), the only meromorphic (complex differentiable) functions are ratios of polynom…

> I wasn't crazy about that line, but basically, sines and cosines make much less sense over say, the rational numbers, whereas polynomials work just fine. They're the most general "nice" function one can define over an arbitrary commutative ring.

This is rather a tautology since polynomials are exactly defined this way. I can imagine that if we built a (fictional) number system rather on properties of the Fourier transformation, sines and cosines would be very natural operations that would have a deep generelization when sufficiently abstracted.

Post reply on HN